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Sharpening the Becker-Stark Inequalities

Abstract

In this paper, we establish a general refinement of the Becker-Stark inequalities by using the power series expansion of the tangent function via Bernoulli numbers and the property of a function involving Riemann's zeta one.

1. Introduction

Steckin [1] (or see Mitrinovic [2, 3.4.19, page 246]) gives us a result as follows.

Theorem 1.1 (see [1, Lemma ]).

If , then

(11)

Later, Becker and Stark [3] (or see Kuang [4, 5.1.102, page 248]) obtain the following two-sided rational approximation for .

Theorem 1.2.

Let , then

(12)

Furthermore, and are the best constants in (1.2).

In fact, we can obtain the following further results.

Theorem 1.3.

Let , then

(13)

Furthermore, and are the best constants in (1.3).

In this paper, in the form of (1.2) and (1.3) we shall show a general refinement of the Becker-Stark inequalities as follows.

Theorem 1.4.

Let , and let be a natural number. Then

(14)

holds, where , and

(15)

where are the even-indexed Bernoulli numbers.

Furthermore, and are the best constants in (1.4).

2. Four Lemmas

Lemma 2.1.

The function is decreasing, where is Riemann's zeta function.

Proof.

is equivalent to the function , which is decreasing.

Lemma 2.2 (see [5, Theorem ]).

Let be Riemann's zeta function and the even-indexed Bernoulli numbers. Then

(21)

Lemma 2.3 (see [6, 1.3.1.4 (1.3)]).

Let . Then

(22)

Lemma 2.4.

Let and . Then , where

(23)

Proof.

By Lemma 2.3, we have

(24)

Since is decreasing by Lemma 2.1, it follows that

(25)

From Lemma 2.2, we get

(26)

which implies that for .

3. Proofs of Theorems

Proof of Theorem 1.4.

Let

(31)

Then

(32)

By Lemma 2.4, we have for , and is decreasing on .

At the same time, = by (3.1), and by (3.2), so and are the best constants in (1.4).

Proof of Theorem 1.3.

Let in Theorem 1.4; we obtain that and . Then the proof of Theorem 1.3 is complete.

References

  1. 1.

    Steckin SB: Some remarks on trigonometric polynomials. Uspekhi Matematicheskikh Nauk 1955, 10(1(63)):159–166.

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  2. 2.

    Mitrinovic DS: Analytic Inequalities. Springer, New York, NY, USA; 1970:xii+400.

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  3. 3.

    Becker M, Strak EL: On a hierarchy of quolynomial inequalities for tanx. University of Beograd Publikacije Elektrotehnicki Fakultet. Serija Matematika i fizika 1978, (602–633):133–138.

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    Kuang JC: Applied Inequalities. 3rd edition. Shangdong Science and Technology Press, Jinan City, China; 2004.

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    Scharlau W, Opolka H: From Fermat to Minkowski, Undergraduate Texts in Mathematics. Springer, New York, NY, USA; 1985:xi+184.

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    Jeffrey A: Handbook of Mathematical Formulas and Integrals. 3rd edition. Elsevier Academic Press, San Diego, Calif, USA; 2004:xxvi+453.

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Correspondence to Ling Zhu.

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Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Zhu, L., Hua, J. Sharpening the Becker-Stark Inequalities. J Inequal Appl 2010, 931275 (2010). https://doi.org/10.1155/2010/931275

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Keywords

  • Power Series
  • Natural Number
  • Series Expansion
  • Zeta Function
  • Power Series Expansion